These notes largely concern autocorrelation—Chapter 12

Issues Using OLS with Time Series Data

Time series data different from cross section in terms of source of variation in x and y—temporal ordering

2nd difference—NOT randomly sampled in same way as cross sectional—each obs not i.i.dWhy?

Data over time is a “stochastic process”—we have one realization of the process from a set of all possible realizations

Leads to a Number of Common problems:

1. Errors correlated over time—high errors today high next time (biased standard errors but not biased coefficients)

2. Effects may take a while to appear difficult to know how long should wait to see effects (tax cuts—is growth in Clinton years due to Clinton? Reagan?) (specification problem)

3. Feedback effects (x y but after seeing y, policy makers adjust x) (specification problem—can lead to biased coeffs)

4. Trending data over time data series can look like they are related, but really is “spurious” (biased coeffs)

Related Issue: Prediction—often want a prediction of future prices, GDP, etc.—Need to use properties of existing series to make that prediction

In time series estimation, have three interrelated modeling issues:

1. What is the correct model specification? Should lagged values of x vars or y vars be included? Should time trends be included? (See Chapter 10 on this + first part of Ch 18)

2. What is the stochastic process? If the data generating process has a unit root (effect of a shock doesn’t decay—e.g., random walk), approach estimation differently. For example, first differencing a common solution. (See Ch 11 on this + Ch 18)

3. Are the errors correlated over time? Not specifying the model appropriately can introduce this problem.

Of the three problems, this is the easiest to deal with and so we will deal with this problem first, others as time permits (See Ch 12 on this)

4.

Note to 2016 Class: we did not discuss this material from Chapter 10. I’m just putting it in here for your reference.

Recall Chapter 10 ModelsThese models dealt with problems 2 and 4 listed above

1. Static model-- Change in z has an immediate effect—in same period—on yyt = 0 + 1zt + ut t=1,2,…n

2. Finite Distributed lag Modelyt = + 0zt + 1zt-1 + 2zt-2 + ut t=1,2,…nKnow number of lags

3. Trending Data:Add a trend yt = 0 + 1t + et , t= 1,2Or Detrend the data

Note that if DO NOT correctly specify the model (e.g., with lagged data), can generate serial correlation. Correct specification is the first problem to address.

What is serial correlation and why is it a problem?

Note: Many authors use serial and auto-correlation interchangeably. Some use auto corr to refer to serial correlation within a series itself and serial correlation to refer to lagged correlation between two time series. I’ll use them interchangeably.

Serial correlation comes when errors from one time period are carried over into future time periods (problem # 1 listed above)

Can also occur spatially—errors in this area are correlated with errors in adjacent area

Positive serial correlation often caused by

o Inertia—some economic time series have “momentum”

o Correlation in omitted variables over time

o Special case of that Model Misspecification—e.g., omitted dynamic terms (lagged dependent or independent variables, trends)

o Correlation in measurement error component of error term

o Data is already interpolated (e.g., data between Census years)

o Theoretical predictions--adaptive expectations, some partial adjustment process

o Non-stationarity of the stochastic process itself—may discuss later

I. Finite Sample Properties of OLS under Classical Assumptions

Have time series analogs to all Gauss Markov assumptionsUse this to identify common problems in time-series data

TS1 Linear in Parameters—ok hereTS2 No perfect collinearity—ok hereTS3 Zero conditional mean

Two ways to express this: Strongest condition: E(ut|X)=0, t=1,2,…n

o error at time t (ut) is uncorrelated with each explanatory variable in EVERY time period

o known as STRICT exogeneityo need this condition for unbiasedness

E(ut, xt1, xt2, . . . . .xtk) = E(ut|xt) = 0o if holds for same period (ut uncorrelated with xt) that is

contemporaneous exogeneityo this assumption is sufficient for consistency

Why would STRICT EXOGENEITY assumption fail?1. As before, omitted vars and measurement error2. Lagged effects of x

look at model yt = 0 + 1zt + ut

ut can’t be correlated with zt, or with past or future zt z can’t have a lagged effect on y (if does, have specified model incorrectly—use distributed lag)BUT As noted above, often DO have effects that emerge over time

3. no feedback of y on future values of z—example of this?(book: murder rate and police)Again, as noted above, often DO have feedback effects

4. Lagged effects of y—will discuss later

TS4 HomoskedasticityTS5 NO serial correlation

Corr(ut,us|X)=0 for all ts

If violated, errors exhibit autocorrelation

Example: AR(1) Process First Order Autoregressive process:

True model: yt = β0+β1x1t + β2x2t + . . . .βkXkt + t

t = t-1 + vt 0≤||≤1[If had 2 lags, would be AR(2)]

vt is the idiosyncratic part of the error, Indep of other errors over time, N(0, 2

v) et is NOT indep of other errors over time, N(0, 2

) error in time t is determined by the diminishing value of error in

previous period () + addition of random variable v, with EV(0)

Implies that error in any period is reflected in all future periods

Var(t) = E(t2) = E[(t-1 + vt)2] = E[22

t-1 + v2t + 2t-1vt]

= 2E(2t-1) + E(v2

t) b/c t-1 and vt are indepVar(t) = 2Var(t) + 2

v if (t) is homoskedastic Var(t)= Var(t-1)

Var(t)= 2v /(1- 2) Note that when =0, no autocorrel.

How are the errors related over time?

Cov(t, t-1) = E(t, t-1)= E[(t-1 + vt) t-1] = E(2t-1 + t-1vt) =

E(2t-1) = Var(t) = 2

Similarly, Cov(t, t-2)= 22 , Cov(t, t-3)= 32

Similarly, Cov(t, t-s)= s2

Note that is the correlation coefficient between errors at time t and t-1. Also known as coefficient of autocorrelation at lag 1

Serial correlation leads to biased standard errors

If y is positively serially correlated and x is positively serially correlated, will understate the errors

Show figure 6.1 for why Note that 1st case have positive error initially, second case have

negative error initially Both cases equally likely to occurunbiased But OLS line fits the data points better than true line

With algebra:

Usual OLS Estimator yt = β0+β1x1t + t

var ( β̂1)= σ2

∑ x i2

With AR(1)

var ( β̂1)= σ2

∑ x t2 [1+2 ρ ∑ x t x t−1

∑ x t2 +2 ρ2 ∑ x t x t−2

∑ x t2 +…+2 ρn−1 ∑ x t xn

∑ x t2 ]

How does this compare with standard errors in OLS case?Depends on sign of p and type of autocorrelation in xs

If x is positively correlated over time and p is positive, OLS will understate true errors

T, F stats all wrong

R2 wrong

See Gujarati for a Monte Carlo experiment on how large these mistakes can be

Brief Digression on Stationarity

Critical that || <1—otherwise these variances and covariances are undefined.

If || <1, we say that the series is stationary. If =1, nonstationary.

Chapter 11 in your book discusses concept of stationarity. For now, brief definition: If mean, variance, and covariance of a series are time invariant, series is stationary.

May discuss later tests of stationarity and what to do if data series is not stationary.

Autocorrelation function will provide a description of correlation over time:

ρk=E[ ( y t−μY )( y t+k−μY ) ]

√ E[ ( y t−μY )2 ] E[ ( y t +k−μY )2 ]=

Cov ( yt , y t+k )σ y t

σ y t+k

If process is stationary,

ρk=E[ ( y t−μY )( y t +k−μY ) ]

σy

2 =γ k

γ 0

In practice, estimate this using sample averages for means

Note that this is symmetricρk =ρ−k

Note that if ρk=0 for all k>0 In that case, yt = t “white noise” No value in using a time series model to forecast the series

Wooldridge: Weak dependency means this function goes to zero “sufficiently quickly” as k

Show Figures 16.3 and 16.4

Note that 16.3 appears to be stationary. Autocorrelation function in 16.4 falls off quickly—can use the function to test for stationarity

Show Figures 16.6—autocorrelation function of a non-stationary seriesDifference until autocorrelation function exhibits this “dropping off” pattern

Wooldridge: Weak dependency means this function goes to zero “sufficiently quickly” as k

Can use Q statistic or Ljung-Box (LB) statistic to examine joint significance that up to certain lag are equal to zero

Q=n∑k=1

m

ρk 2 (should have hat) m is lag length~chi squared m

Examples of stationary, weakly dependent series:

Stable AR(1) yt = 1yt-1 + t where |1| < 1

Wooldridge shows that this process has finite variance, correlation gets smaller and smaller as lags get large

MA(1) xt = et + 1et-1 t=1,2,. . . .T

Note that adjacent terms are correlated.However, if are 2 or more time periods apart, are independent

Tests for Serial Correlation

1. Graphical method

Graph (residuals) errors in the equation---very commonly done.

Can also plot residuals against lagged residuals—see Gujarati fig 12.9

2. Durbin Watson Test Oldest test for serial correlationP&R goes through extension when have lagged y’s in model—see 6.2.3 for details

Null hypothesis: No serial correlation =0Alternative: 0 (two tailed)

>0 (one tailed)

Test statistic:Step 1: Run OLS model yt = β0+β1x1t + β2x2t + . . . .βkXkt + t

Step 2: Calculate predicted residualsStep 3: Form test statistic

DW=∑t=2

T

( ε̂t− ε̂t−1)2

∑t=1

T

( ε̂ t )2

≃2(1− ρ̂ )

(See Gujarati pg 435 to derive)

Assumptions:1. Regression includes intercept term2. Xs are fixed in repeated sampling—non-stochastic (problematic in

time series context)3. Can only be used for 1st order autoregression processes4. Errors are normally distributed5. No lagged dependent variables—not applicable in those models6. No missing obs

This statistic ranges from 0 to 4 ε̂ t are close to each other Positive serial correlationDW will be

close to zero (below 2) No serial correlation DW will be close to 2 Negative serial correlation DW will be large (above 2)

Exact interpretation difficult because sequence of predicted error terms depends on x’s as well if x’s are serially correlated, correlation of predicted errors may be related to this and not serial correlation of s

2 critical values dL and dU

--see book for chart

STATA: estat dwstat

3. Breusch-Godfrey testThis is yet another example of an LM test

Null hypothesis: Errors are serially independent up to order p

One X:

Step 1: Run OLS model yt = β0+β1x1t + t

(Regression run under the null)

Step 2: Calculate predicted residuals

Step 3: Run auxiliary regression ε̂ t=α 1+α 2X + ρ ε̂t−1+vt

Step 4: T-test on ρ̂

STATA: estat bgodfrey, lags(**)

Multiple X, multiple lags

Step 1: Run OLS model yt = β0+β1x1t + β2x2t + . . . .βkXkt + t

(Regression run under the null)

Step 2: Calculate predicted residuals

Step 3: Run auxiliary regression ε̂ t=α 1+α 2 Xs+ρ1 ε̂ t−1+ ρ2 ε̂t−1+. .. . ++ ρp ε̂ t−p+v t

with higher order lags—Bruesch-Godfrey test

Step 4: (n-p)R2 ~ χ2(p)

BP test is more general than DW test—cam include laggesd Ys, moving average models

Do need to know p—order or the lag. Will talk some about this choice later.

Correcting for Serial Correlation

1. Check—is it model misspecification?--trend variable?--quadratics?--lagged variables?

2. Use GLS estimator—see below

3. Use Newey –West standard errors—like robust standard errors

Approach 1: GLS Estimators

Correction1: Known : Adjust OLS regression to get efficient parameter estimates

Want to transform the model so that errors are independent t = t-1 + vt want to get rid of t-1 part

How? Linear model holds for all time periods.yt-1 = β0+β1x1t-1 + β2x2t-1 + . . . .βkXkt-1 + t-1

1. Multiply above by 2. Subtract from base model:

y*t = β0(1-) + β1x*1t + β2x*2t + . . . .βkX*kt + vt

Where y*t = yt - yt-1 , same for xs

Note that this is like a first difference, only are subtracting part and not whole of yt-1Generalized differences

Now error has a mean =0 and a constant variance

Apply OLS to this transformed model efficient estimates

This is the BLUE estimator

PROBLEM: don’t know

Correction2: Don’t Know --Cochrane-Orcutt

Idea: start with a guess of and iterate to make better and better guesses

Step 1: Run ols on original modelyt = β0+β1x1t + β2x2t + . . . .βkXkt + t

Step 2: Obtain predicted residuals and run following regressionε̂ t=ρ ε̂t−1+v t

Step 3: Obtain predicted value of . Transform data using

generalized differencing transformation y t¿= y t− ρ̂ y t−1 , same for X*

Step 4: Rerun regression using transformed datay t

¿=β0(1− ρ̂)+β1 x1t¿ +.. .+βk xkt

¿ +v t

Obtain new estimates of betas--^̂β

Step 5: Form new estimated residuals using newly estimated betas and ORIGINAL data (not transformed data)

^̂ε t= yt−( ^̂β0+^̂β1 x1 t+. . .+ ^̂βk xkt

Iterate until new estimates of are “close” to old estimates (differ by .01 or .005)

Correction3: Don’t Know --Hildreth-Lu (less popular)

Numerical minimization methodMinimize sum of squared residuals for various guesses of for

y*t = β0(1-) + β1x*1t + β2x*2t + . . . .βkX*kt + vt

Choose range of potential (e.g., 0, .1, .2, .3, . . . ., 1.0), identify best one (e.g., .3), pick other numbers close by (e.g., .25, .26, . . . , .35), iterate

Correction 4: First difference Model

lies between 0 and 1. Could run a first differenced model as the other extreme. This is the appropriate correction when series is non-stationary—talk about next time.

Recall: Correcting for Serial Correlation

1. Check—is it model misspecification?--trend variable?--quadratics?--lagged variables?

2. Use GLS estimator—see below

3. Use Newey –West standard errors—like robust standard errors

Approach 2: Newey –West standard errors

Extension of White standard errors for heteroskedasticityOnly valid in large samples

Comparison of Approaches: Should you use OLS or FGLS or Newey-West errors?

OLS:--unbiased--consistent--asymptotically normal--t,F, r2 not appropriate

FGLS/Newey West--efficient--small sample properties not well documented—not unbiased--in small samples, then, might be worse--Griliches and Rao rule of thumb—is sample is small (<20, iffy 20-50) and <.3, OLS better than FGLS